Properties of Multiplication

In the last section, we learned how to add long strings of numbers using the
properties of addition. Similarly, it is sometimes necessary to multiply long
strings of numbers without a calculator; this task is made easier by learning
some of the properties of multiplication.

Multiplication and addition have some similar properties. Like addition,
multiplication has a Commutative Property and an Associative Property.

Commutative Property

The commutative property for multiplication states that for any numbers
a and b, the following is always true:

a×b = b×a

For example, 3×4 = 4×3. We can see that this is true because
3×4 = 12 and 4×3 = 12, so 3×4 = 4×3. Just as in addition,
we can multiply a long string of numbers in any order. This can make multiplication
without a calculator easier. For example:

4×6×5 = 4×5×6 = 20×6 = 120

Associative Property

The associative property for multiplication states that for any numbers
a, b, and c, the following is always true:

(a×b)×c = a×(b×c)

For example, (2×5)×6 = 2×(5×6). We can see
that this is true because (2×5)×6 = 10×6 = 60, and
2×(5×6) = 2×30 = 60. Thus,
(2×5)×6 = 2×(5×6)

Identity Property

Multiplication also has its own Identity Property. This property
states that when any number is multiplied by 1, it does not change its identity.
For any number a, the following is always true:

a×1

=

a

1×a

=

a

For example, 45×1 = 45. 1×123 = 123.

We can remember these three properties of multiplication just as we can remember
the corresponding properties of addition. With the Commutative Property of
Multiplication, when only multiplication is involved, numbers can move
("commute") to anywhere in the expression. With the Associative Property of
Multiplication, any numbers that are being multiplied together can "associate"
with each other. Also, multiplying by 1 does not change the Identity of a
number.